As an Immediate Benefit (p. 221) an XP can be used to reroll a dice (and get the best result) or to avoid a GM intrusion.
The rulebook states that with a natural 1 on a d20 roll the GM makes a free intrusion (p. 193).
While the player can pay 1 XP to refuse the (free) GM intrusion, does rerolling the natural 1 have the implicit effect of avoid the intrusion in the first place? Or the intrusion stay and the player must eventually pay 2 XP, one to avoid the intrusion and one to roll again?
Does rerolling the natural 1 have the implicit effect of avoid the intrusion in the first place?
Yes. When you pay XP to reroll a check, you take the better result. Since you're taking the best result out of the dice you've rerolled, you can avoid a 1, and therefore avoid the free GM Intrusion.
Example: I've rolled a 1, and decided to reroll to avoid the GM Intrusion. I pay an XP, reroll and get a 20. I use the 20.
Example: I've rolled a 1. I pay an XP, reroll and get another 1. The "best" roll in both cases is a 1. I must take it (unless I want to pay another XP), and also accept the free GM Intrusion.
Admittedly, the cypher system ruleset is confusing on this point. The verbiage of the rules cited in ShadowKras's answer state that you keep the best roll, but also suggests that you can deny a free GM Intrusion by spending 1XP. Why would you spend 1XP to deny the intrusion if you could spend 1XP to reroll the die and (hopefully) get a better result? These two rules conflict in both spirit and letter, hence your understandable confusion. I haven't found any errata that clarifies the matter, but they're coming out with a second edition which may better word some of the more confusing rules.
FWIW, we've always played it that if you've paid XP to reroll, you keep the best result, and don't have to deal with the consequences of the other rolls. That's the benefit implicit in the rule of paying XP to reroll. Denying intrusions IMO, is best saved for non-free GM intrusions, when you're negotiating with the GM on how they can screw your character.
The free intrusion happens every time a player rolls a 1 on the d20, rerolling would allow you to try succeeding once more, while not rerolling would not only trigger the GM Intrusion but also fail the test. The "free" gm intrusion is one of the special results of a check:
SPECIAL ROLLS
If a character rolls a natural 1, 17, 18, 19, or 20 (meaning the d20 shows that number), special rules come into play. These are explained in more detail in the following sections.
1: Intrusion. The GM makes a free intrusion (see below) and doesn’t award experience points (XP) for it.
17: Damage Bonus. If the roll was an attack, it deals 1 additional point of damage.
18: Damage Bonus. If the roll was an attack, it deals 2 additional points of damage.
19: Minor Effect. If the roll was an attack, it deals 3 additional points of damage. If the roll was something other than an attack, the PC gets a minor effect in addition to the normal results of the task.
20: Major Effect. If the roll was an attack, it deals 4 additional points of damage. If the roll was something other than an attack, the PC gets a major effect in addition to the normal results of the task. If the PC spent points from a stat Pool on the action, the point cost for the action decreases to 0, meaning the character regains those points as if she had not spent them at all.
And when you spend an experience point to reroll a check, you take the best of the two results. If you took the best result, your result was never 1 (unless you roll another 1), and thus the GM cannot make a gm intrusion for free:
Immediate Benefits
The easiest, most straightforward way for a player to use XP is to reroll any roll in the game—even one that she didn’t make herself. This costs 1 XP per reroll. The player can reroll and use either the new roll or the original, whichever is better. She can continue to spend XP on more rerolls, but this can quickly become an expensive proposition. It’s a fine way to attempt to prevent a disaster, but it’s not a good idea to use of a lot of XP to reroll a single action over and over.
As mentioned earlier, a player can also spend 1 XP to refuse a GM intrusion.
If ruled otherwise, you could also be in a situation where it is possible to roll 17-20 two times and take both special roll results. If you have a lot of XP to spend, I don't see how that isn't always a good deal, as you still get to keep one good result.
The Cypher System Rulebook contains a side bar that indicates that any other treatment of the situation than treating the number of the second dice the same way as if it was the initial rule would be a house rule.
Cypher System Rulebook, p. 410 (side "bar", emphasis mine):
This might not be true of your players, but many players rarely, if ever, spend XP to refuse an intrusion from the GM, though they regularly use XP to avoid an intrusion that comes from a bad roll. And there’s nothing wrong with that. Some GMs might want to forbid using an XP to reroll a 1, but there's really no point - if you've got an idea for a good intrusion, you don't need to wait until a player rolls a 1 to use it
